
By corollary 2.2.2 in [1]} \((N/k) U_{k, N}\xrightarrow{} 1\) .Corollary 2.2.2 in [1]} asserts that \((k/N) Y_{N-k, N}\xrightarrow{}1\)  where \(Y_{N-k, N}\)  are order statistics of \({Y_1, \dots , Y_N}\) , and \(P(Y_i\le y) = 1-1/y\)  if \(y\ge 1\)  and zero otherwise. Observe that \(U_{k, N}\overset{d}{=} 1/Y_{N-k, N}\) , the result then follows. See also the proof of theorem  REF. Then the \(F^{-1}_{Fr, \kappa }\)  terms decay if \(k\)  is chosen such that
\(\sqrt{k}\left[ \left(\dfrac{N^\rho +1}{N^\rho }\right)^{1/\kappa } - 1 \right] \rightarrow 0.\) 
